{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:U5IGGM6PMTY7NZQYJM7T3SS7PY","short_pith_number":"pith:U5IGGM6P","canonical_record":{"source":{"id":"1502.03117","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-02-10T21:08:48Z","cross_cats_sorted":[],"title_canon_sha256":"4f7ce4e7cfe2f1dbe279bcfab3d15e3b7b571fed27887977589565a41356d0fa","abstract_canon_sha256":"0581b90e433ce22e5267bb7d990c887e737c1e49ffe36102ca83b66cdbdcca75"},"schema_version":"1.0"},"canonical_sha256":"a7506333cf64f1f6e6184b3f3dca5f7e29f1c9d7f07b76e8579e335075add601","source":{"kind":"arxiv","id":"1502.03117","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.03117","created_at":"2026-05-18T02:27:20Z"},{"alias_kind":"arxiv_version","alias_value":"1502.03117v1","created_at":"2026-05-18T02:27:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.03117","created_at":"2026-05-18T02:27:20Z"},{"alias_kind":"pith_short_12","alias_value":"U5IGGM6PMTY7","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"U5IGGM6PMTY7NZQY","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"U5IGGM6P","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:U5IGGM6PMTY7NZQYJM7T3SS7PY","target":"record","payload":{"canonical_record":{"source":{"id":"1502.03117","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-02-10T21:08:48Z","cross_cats_sorted":[],"title_canon_sha256":"4f7ce4e7cfe2f1dbe279bcfab3d15e3b7b571fed27887977589565a41356d0fa","abstract_canon_sha256":"0581b90e433ce22e5267bb7d990c887e737c1e49ffe36102ca83b66cdbdcca75"},"schema_version":"1.0"},"canonical_sha256":"a7506333cf64f1f6e6184b3f3dca5f7e29f1c9d7f07b76e8579e335075add601","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:20.534735Z","signature_b64":"9K3r6jEFa5WI42WZTkI7KvcqdeHQtNgNqekTRNFPJJ7nc/aE4GhnJ315oOwzjLGODy5yxfrZr4NQ1tpioxy3Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7506333cf64f1f6e6184b3f3dca5f7e29f1c9d7f07b76e8579e335075add601","last_reissued_at":"2026-05-18T02:27:20.533951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:20.533951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1502.03117","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:27:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"N+7SMVIzxKjWIiXvBaKjhRbDJNWhlD9ySCbpD+Ukpn1+zfZDWMaT2pE2OMN1DrlIo8HtSX0fUe32i+Z0DPlFCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T12:23:59.296710Z"},"content_sha256":"9350d337d2c57d5594e15c0180c4eb8b45602267ccd186f6e20822b99c9b47f7","schema_version":"1.0","event_id":"sha256:9350d337d2c57d5594e15c0180c4eb8b45602267ccd186f6e20822b99c9b47f7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:U5IGGM6PMTY7NZQYJM7T3SS7PY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kolmogorov widths and low-rank approximations of parametric elliptic PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Albert Cohen, Markus Bachmayr","submitted_at":"2015-02-10T21:08:48Z","abstract_excerpt":"Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $n$-widths can be controlled by that of the error achieved by best $n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03117","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:27:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZFGqEanZIX9TlG/RfQxk1o84ohi5rcvwWLOPQ8A5PN7ZrkeB9z6IDHrwFz3dVz/kWoCDLYIdovYhrP/6D5qvBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T12:23:59.297434Z"},"content_sha256":"1c185f5199d7cff5ea2c87c63d0ca773542399b70cf7394df813620382b3ece3","schema_version":"1.0","event_id":"sha256:1c185f5199d7cff5ea2c87c63d0ca773542399b70cf7394df813620382b3ece3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U5IGGM6PMTY7NZQYJM7T3SS7PY/bundle.json","state_url":"https://pith.science/pith/U5IGGM6PMTY7NZQYJM7T3SS7PY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U5IGGM6PMTY7NZQYJM7T3SS7PY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T12:23:59Z","links":{"resolver":"https://pith.science/pith/U5IGGM6PMTY7NZQYJM7T3SS7PY","bundle":"https://pith.science/pith/U5IGGM6PMTY7NZQYJM7T3SS7PY/bundle.json","state":"https://pith.science/pith/U5IGGM6PMTY7NZQYJM7T3SS7PY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U5IGGM6PMTY7NZQYJM7T3SS7PY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:U5IGGM6PMTY7NZQYJM7T3SS7PY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0581b90e433ce22e5267bb7d990c887e737c1e49ffe36102ca83b66cdbdcca75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-02-10T21:08:48Z","title_canon_sha256":"4f7ce4e7cfe2f1dbe279bcfab3d15e3b7b571fed27887977589565a41356d0fa"},"schema_version":"1.0","source":{"id":"1502.03117","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1502.03117","created_at":"2026-05-18T02:27:20Z"},{"alias_kind":"arxiv_version","alias_value":"1502.03117v1","created_at":"2026-05-18T02:27:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1502.03117","created_at":"2026-05-18T02:27:20Z"},{"alias_kind":"pith_short_12","alias_value":"U5IGGM6PMTY7","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"U5IGGM6PMTY7NZQY","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"U5IGGM6P","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:1c185f5199d7cff5ea2c87c63d0ca773542399b70cf7394df813620382b3ece3","target":"graph","created_at":"2026-05-18T02:27:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $n$-widths can be controlled by that of the error achieved by best $n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD m","authors_text":"Albert Cohen, Markus Bachmayr","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-02-10T21:08:48Z","title":"Kolmogorov widths and low-rank approximations of parametric elliptic PDEs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.03117","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:9350d337d2c57d5594e15c0180c4eb8b45602267ccd186f6e20822b99c9b47f7","target":"record","created_at":"2026-05-18T02:27:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0581b90e433ce22e5267bb7d990c887e737c1e49ffe36102ca83b66cdbdcca75","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NA","submitted_at":"2015-02-10T21:08:48Z","title_canon_sha256":"4f7ce4e7cfe2f1dbe279bcfab3d15e3b7b571fed27887977589565a41356d0fa"},"schema_version":"1.0","source":{"id":"1502.03117","kind":"arxiv","version":1}},"canonical_sha256":"a7506333cf64f1f6e6184b3f3dca5f7e29f1c9d7f07b76e8579e335075add601","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7506333cf64f1f6e6184b3f3dca5f7e29f1c9d7f07b76e8579e335075add601","first_computed_at":"2026-05-18T02:27:20.533951Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:27:20.533951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9K3r6jEFa5WI42WZTkI7KvcqdeHQtNgNqekTRNFPJJ7nc/aE4GhnJ315oOwzjLGODy5yxfrZr4NQ1tpioxy3Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T02:27:20.534735Z","signed_message":"canonical_sha256_bytes"},"source_id":"1502.03117","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9350d337d2c57d5594e15c0180c4eb8b45602267ccd186f6e20822b99c9b47f7","sha256:1c185f5199d7cff5ea2c87c63d0ca773542399b70cf7394df813620382b3ece3"],"state_sha256":"ffe449d2074aa9fddb66d8592298b9bfefa43728b68ca534db7764f5b08ce33b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6RebAoidRBJyfLHlbNWGzQVkpwwmwI4xZetG3VFDdMzX+b2UuRx15VWVCa0jmzmAdjcwXAU/N6nqpIYlMf4yAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T12:23:59.301651Z","bundle_sha256":"7c4a9b6fa425c10908abb0bdedfdb739577fe8a0f93f213959759229b88e8cd5"}}